The genus of plants which comprise edible mints are known as Mentha, and they belong to the Lamiaceae Family. There are many plants which belong in the same Family (Lamiaceae) such as your 'catmint', but they are not necessarily edible, although people might commonly refer to them as mint or mint family (even though 'mint family' does not exist). 'Catmint' is more usually referred to as catnip, and is a common name for Nepeta faassenii; it is not used for culinary purposes. Monarda (also a member of the Lamiaceae, common name Bergamot or Bee Balm) is sometimes used in food or as a medicine, but it does not have a minty taste because it is not a Mentha. The family Lamiaceae also includes, for instance, Lamiums, such as Lamium purpureum - they are edible, but again, are not Mentha and so are not minty tasting and are usually used as ornamental plants in the garden.
It is important that you know the Latin or botanical names for your plants rather than just the common names, or you might end up eating something you believe is mint that isn't mint at all, or which might actually be toxic.
In regard to varieties of Mentha, there is much argument botanically about the naming of various mint plants, see here https://en.m.wikipedia.org/wiki/Mentha. As you can see from the link, many are named (some common names for named varieties are chocolate mint, lemon mint, spearmint and so on) but there is still confusion about different varieties.
In terms of growing mints, they rapidly outgrow a pot - their nature is to put out long rhizomes under the soil, and to pop up a foot or three away from the site of the original plant. Obviously, this is not possible in a pot, and so the plant struggles within a few months to produce any useable leaves. That necessitates a lot of unpotting, dividing and repotting of the sections in order to keep up supplies of mint leaves for culinary use.
If you want to plant mint direct into the ground, then a root rhizome barrier inserted to a depth of 18 inches around each plant should prevent it spreading throughout the garden, but each plant should be given plenty of space within the enclosing barrier, say 2 or 3 feet square. Even so, after a couple of years, it will be necessary to dig up the plant, discard the unproductive centre, and replant the outer parts which are still productive.
You may be told that mint is not invasive, but having had the unlovely experience myself many times of having to deal with mint left to run riot through a garden, I can absolutely testify that it is extremely invasive. A lawn with mint growing in it might be fragrant to walk on, but it does not make for a good lawn...
As for your plants making it through winter, if your winters are severe, there is a risk that the small pots might freeze, and that will kill all your plants. Temperatures below 32 degrees F, or zero degrees C night AND day for a week or longer means they will freeze. Clustering the pots together in the warmest, most sheltered spot you have, and surrounding them with insulating material (straw bales or something) might prevent that happening, but it does depend how cold your winters actually get. Alternatively, removing them to a greenhouse over winter if you can will get them through - they are hardy plants, but much more vulnerable through the roots in pots. The plant or cutting you have which has a root but not yet a rhizome, provided you pot it up and it starts growing, will have no more trouble coming through the winter than your other plants, so long as the pot does not freeze.
